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Fractal Fract
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Feature Papers for Numerical and Computational Methods Section
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Feature Papers for Numerical and Computational Methods Section

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (25 June 2024) | Viewed by 18968

Special Issue Editor

Prof. Dr. Riccardo Caponetto

E-Mail Website
Guest Editor
Department of Engineering, University of Messina, 98158 Messina, Italy
Interests: systems modeling and control; fractional order systems; soft computing techniques
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

As Section Editor-in-Chief of “Numerical and Computational Methods” in Fractal and Fractional, I am pleased to announce the new Special Issue “Feature Papers for Section 'Numerical and Computational Methods'”. This Special Issue is designed to publish high-quality papers in Fractal and Fractional. We welcome submissions from Editorial Board Members and outstanding scholars invited by the Editorial Board and by the Editorial Office. The scope of this Special Issue includes, but is not limited to, all aspects of the numerical approach to solving fractal and fractional problems.

You are welcome to send short proposals to our Editorial Office ([email protected]). They will firstly be evaluated by Academic Editors. Selected papers will be thoroughly and rigorously peer reviewed before publication.

Prof. Dr. Riccardo Caponetto
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus
  • numerical methods
  • approximation methods
  • computational procedures
  • algorithms
  • digital implementation
  • hardware in the loop implementation
  • FPGA implementation
  • data mining with fractional calculus methods
  • fractional calculus with artificial intelligence applications
  • image/signal analyses based on fractional calculus
  • fuzzy fractional calculus
  • neural computations with fractional calculus
  • applications of fractional calculus in nonlinear science
  • applications in Control, Mechanics, Financial Mathematics, Engineering, Biomedicine, etc

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Related Special Issue

Published Papers (13 papers)

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Research

30 pages, 2530 KiB  
Article
Stationary Responses of Seven Classes of Fractional Vibrations Driven by Sinusoidal Force
by Ming Li
Fractal Fract. 2024, 8(8), 479; https://doi.org/10.3390/fractalfract8080479 - 16 Aug 2024
Cited by 1 | Viewed by 547
Abstract
This paper gives the contributions in three folds. First, we propose fractional phasor motion equations of seven classes of fractional vibrators. Second, we put forward fractional phasor responses to seven classes of fractional vibrators. Third, we bring forward the analytical expressions of stationary [...] Read more.
This paper gives the contributions in three folds. First, we propose fractional phasor motion equations of seven classes of fractional vibrators. Second, we put forward fractional phasor responses to seven classes of fractional vibrators. Third, we bring forward the analytical expressions of stationary responses in time to seven classes of fractional vibration systems driven by sinusoidal force using elementary functions. The present results show that there are obvious effects of fractional orders on the sinusoidal stationary responses to fractional vibrations. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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18 pages, 590 KiB  
Article
New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems
by Hany Mostafa Ahmed
Fractal Fract. 2024, 8(4), 199; https://doi.org/10.3390/fractalfract8040199 - 29 Mar 2024
Cited by 1 | Viewed by 974
Abstract
In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs [...] Read more.
In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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27 pages, 2722 KiB  
Article
A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes
by Mudassir Shams, Naila Rafiq, Bruno Carpentieri and Nazir Ahmad Mir
Fractal Fract. 2024, 8(3), 162; https://doi.org/10.3390/fractalfract8030162 - 12 Mar 2024
Cited by 2 | Viewed by 1312
Abstract
In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous [...] Read more.
In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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26 pages, 22673 KiB  
Article
New Generalized Jacobi Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Multi-Term Variable-Order Time-Fractional Diffusion-Wave Equations
by Hany Mostafa Ahmed
Fractal Fract. 2024, 8(1), 68; https://doi.org/10.3390/fractalfract8010068 - 18 Jan 2024
Cited by 6 | Viewed by 1365
Abstract
The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing [...] Read more.
The current study discusses a novel approach for numerically solving MTVO-TFDWEs under various conditions, such as IBCs and DBCs. It uses a class of GSJPs that satisfy the given conditions (IBCs or DBCs). One of the important parts of our method is establishing OMs for Ods and VOFDs of GSJPs. The second part is using the SCM by utilizing these OMs. This algorithm enables the extraction of precision and efficacy in numerical solutions. We provide theoretical assurances of the treatment’s efficacy by validating its convergent and error investigations. Four examples are offered to clarify the approach’s practicability and precision; in each one, the IBCs and DBCs are considered. The findings are compared to those of preceding studies, verifying that our treatment is more effective and precise than that of its competitors. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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23 pages, 429 KiB  
Article
An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations
by Xuehui Ren, Siriguleng He and Hong Li
Fractal Fract. 2023, 7(10), 757; https://doi.org/10.3390/fractalfract7100757 - 13 Oct 2023
Cited by 1 | Viewed by 1121
Abstract
In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. [...] Read more.
In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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28 pages, 7528 KiB  
Article
Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM
by Mamta Kapoor, Nasser Bin Turki and Nehad Ali Shah
Fractal Fract. 2023, 7(9), 650; https://doi.org/10.3390/fractalfract7090650 - 25 Aug 2023
Viewed by 989
Abstract
In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after [...] Read more.
In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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20 pages, 62898 KiB  
Article
An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving
by Jian Sun, Ling Wang and Dianxuan Gong
Fractal Fract. 2023, 7(6), 448; https://doi.org/10.3390/fractalfract7060448 - 31 May 2023
Cited by 2 | Viewed by 1424
Abstract
The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely [...] Read more.
The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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14 pages, 2830 KiB  
Article
Using Particle Swarm Optimization and Artificial Intelligence to Select the Appropriate Characteristics to Determine Volume Fraction in Two-Phase Flows
by Abdullah M. Iliyasu, Abdallah S. Benselama, Dakhkilgova Kamila Bagaudinovna, Gholam Hossein Roshani and Ahmed S. Salama
Fractal Fract. 2023, 7(4), 283; https://doi.org/10.3390/fractalfract7040283 - 24 Mar 2023
Cited by 7 | Viewed by 1647
Abstract
Global demand for fossil fuels has increased the importance of flow measurement in the oil sector. As a result, a new submarket in the flowmeter business has opened up. To improve the accuracy of gamma-based two-phase flowmeters, this study employs time-feature extraction methods, [...] Read more.
Global demand for fossil fuels has increased the importance of flow measurement in the oil sector. As a result, a new submarket in the flowmeter business has opened up. To improve the accuracy of gamma-based two-phase flowmeters, this study employs time-feature extraction methods, a particle swarm optimization (PSO) based feature selection system, and an artificial neural network. This article proposes a fraction detection system that uses a 137Cs gamma source, two NaI detectors for recording the photons, and a Pyrex-glass pipe between them. The Monte Carlo N Particle method was used to simulate the geometry mentioned above. Thirteen time-domain features were extracted from the raw data recorded by both detectors. Optimal characteristics were identified with the help of PSO. This procedure resulted in the identification of eight efficient features. The input-output relationship was approximated using a Multi-Layer Perceptron (MLP) neural network. The innovation of the present research is in the use of a feature extraction technique based on the PSO algorithm to determine volume percentages, with results such as: (1) introducing eight appropriate time characteristics in determining volume percentages; (2) achieving an accuracy of less than 0.37 in root mean square error (RMSE) and 0.14 in mean square error (MSE) while predicting the volume fraction of components in a gas-liquid two-phase flow; and (3) reducing the calculation load. Utilizing optimization-based feature selection techniques has allowed for the selection of meaningful inputs, which has decreased the volume of computations while boosting the precision of the presented system. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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19 pages, 4274 KiB  
Article
Numerical Simulation of Nonlinear Dynamics of Breast Cancer Models Using Continuous Block Implicit Hybrid Methods
by Dauda Gulibur Yakubu, Abdulhameed Mohammed, Adamu Garba Tahiru, Kadas Saidu Abubakar and Magaji Yunbunga Adamu
Fractal Fract. 2023, 7(3), 237; https://doi.org/10.3390/fractalfract7030237 - 7 Mar 2023
Cited by 1 | Viewed by 2447
Abstract
In the search for causes and cures of cancer diseases, many mathematical models developed have resulted in systems of nonlinear stiff ordinary differential equations. With these models, many numerical estimates of biological knowledge of the parameters have been obtained, a number of phenomena [...] Read more.
In the search for causes and cures of cancer diseases, many mathematical models developed have resulted in systems of nonlinear stiff ordinary differential equations. With these models, many numerical estimates of biological knowledge of the parameters have been obtained, a number of phenomena interpreted, and predictions were made in order to gain further knowledge of cancer development and possible treatment. In this study, numerical simulations of the models were performed using continuous block implicit hybrid methods and the results obtained support the theoretical and clinical findings. We analyzed the interactions among the various tumor cell populations and present the results graphically. From the graphical representation of results, one can clearly see the effects of all the tumor cell populations involved in the competition, as well as the effects of some treatments by the applications of some therapeutic agents which have been heavily used in the clinical treatments of breast cancer. The treatments in the past were mostly conventional chemotherapies, which were used either singly (alone) or in combination with each other or other therapies, and all played vital roles, except for the side effects that these therapies incur in normal tissues and organs. Thus, from recent research works, it is now clear that in many cases they do not represent a complete cure. Therefore, the need to address not only the preventative measures of breast cancer, but also more successful treatment, is clear, and can be successfully achieved to increase the survival rate of breast cancer patients. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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12 pages, 317 KiB  
Article
Existence Results for Nonlinear Fractional Differential Inclusions via q-ROF Fixed Point
by Lariab Shahid, Maliha Rashid, Akbar Azam and Faryad Ali
Fractal Fract. 2023, 7(1), 41; https://doi.org/10.3390/fractalfract7010041 - 30 Dec 2022
Cited by 1 | Viewed by 1253
Abstract
Fractional Differential inclusions, the multivalued version of fractional differential equations, yellow play a vital role in various fields of applied sciences. In the present article, a class of q-rung orthopair fuzzy (q-ROF) set valued mappings along with q-ROF upper/lower semi-continuity [...] Read more.
Fractional Differential inclusions, the multivalued version of fractional differential equations, yellow play a vital role in various fields of applied sciences. In the present article, a class of q-rung orthopair fuzzy (q-ROF) set valued mappings along with q-ROF upper/lower semi-continuity have been introduced. Based on these ideas, existence theorems for a numerical solution of a distinct class of fractional differential inclusions have been achieved with the help of Schaefer type and Banach contraction fixed point theorems. A physical example is also provided to validate the hypothesis of the main results. The notion of q-rung orthopair fuzzy mappings along with the use of fixed point techniques and a new-fangled Caputo type fractional derivative are the principal novelty of this article. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
18 pages, 3914 KiB  
Article
The First Integral of the Dissipative Nonlinear Schrödinger Equation with Nucci’s Direct Method and Explicit Wave Profile Formation
by Muhammad Abu Bakar, Saud Owyed, Waqas Ali Faridi, Magda Abd El-Rahman and Mohammed Sallah
Fractal Fract. 2023, 7(1), 38; https://doi.org/10.3390/fractalfract7010038 - 29 Dec 2022
Cited by 12 | Viewed by 1648
Abstract
The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1+1)− dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci’s scheme [...] Read more.
The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1+1)− dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci’s scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci’s direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci’s scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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22 pages, 854 KiB  
Article
Analytical and Computational Problems Related to Fractional Gaussian Noise
by Yuliya Mishura, Kostiantyn Ralchenko and René L. Schilling
Fractal Fract. 2022, 6(11), 620; https://doi.org/10.3390/fractalfract6110620 - 22 Oct 2022
Cited by 2 | Viewed by 1345
Abstract
We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of [...] Read more.
We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection. In particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for H>1/2, and, in particular, monotone, convex, log-convex along with further useful properties. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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19 pages, 810 KiB  
Article
Modal Shifted Fifth-Kind Chebyshev Tau Integral Approach for Solving Heat Conduction Equation
by Ahmed Gamal Atta, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid and Youssri Hassan Youssri
Fractal Fract. 2022, 6(11), 619; https://doi.org/10.3390/fractalfract6110619 - 22 Oct 2022
Cited by 11 | Viewed by 1465
Abstract
In this study, a spectral tau solution to the heat conduction equation is introduced. As basis functions, the orthogonal polynomials, namely, the shifted fifth-kind Chebyshev polynomials (5CPs), are used. The proposed method’s derivation is based on solving the integral equation that [...] Read more.
In this study, a spectral tau solution to the heat conduction equation is introduced. As basis functions, the orthogonal polynomials, namely, the shifted fifth-kind Chebyshev polynomials (5CPs), are used. The proposed method’s derivation is based on solving the integral equation that corresponds to the original problem. The tau approach and some theoretical findings serve to transform the problem with its underlying conditions into a suitable system of equations that can be successfully solved by the Gaussian elimination method. For the applicability and precision of our suggested algorithm, some numerical examples are given. Full article
(This article belongs to the Special Issue Feature Papers for Numerical and Computational Methods Section)
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